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  • Level: GCSE
  • Subject: Maths
  • Word count: 3695

In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.

Extracts from this document...

Introduction

Daniel Lovegrove                        10CW

Maths Coursework

MATHS

NumberGridCoursework

Introduction

In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.

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such as

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36

45

46


when I find the product of the 2 sets of opposite corners I also find that the difference between these answers is ten:

       2x2 squares

              square                        products                 difference

35

36

45

46

45 x 36 = 1620

35 x 46 = 1610

10

57

58

67

68

57 x 68 = 3876

67 x 58 = 3886

10

85

86

95

96

85 x 96 = 8160

95 x 86 = 8170

10

My findings are laid out here algebraically (s= starting number)

35

36

45

46

The +10 in the lower left corner is there because every row has ten numbers and so if you look down the columns you will see that

here 35+10=40 and 36+10=46 and the

+1 is there obviously because every

number you count to the right is bigger

by 1 i.e. 35+1=36 I shall use the same method in all my working throughout this project

S

S+1

S+10

S+11

S(S+11) = S2 +11S

(S+10)(S+1) = S2 +10S +S + 10 = S2 + 11S + 10

The difference is ten

To find this I multiplied the corners of my square and found the difference between the products.

Next I investigated 3 x 3 squares within a 10 x 10 grid and found that the difference was 40

      Square               products             difference

1

2

3

11

12

13

21

22

23

1 x 23 = 23

21 x 3 = 63

40

55

56

57

65

66

67

75

76

77

55 x 77 = 4235

75 x 57 = 4275

40

44

45

46

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55

56

64

65

66

44 x 66 = 2904

64 x 46 = 2944

40

Again my results are laid out algebraically (s = starting number)

S

S+1

S+2

S+10

S+11

S+12

S+20

S+21

S+22

S(S+22)

...read more.

Middle

S(S+10) = S2+10S

(S+9)(S+1) = S2+10s+9

The difference is nine

My investigation continues with 3x3 squares

Square

Products

Difference

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30

31

38

39

40

47

48

49

29x49=1421

47x31=1457

36

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14

15

22

23

24

31

32

33

13x33=429

31x15=465

36

48

49

50

57

58

59

66

67

68

48x68=3264

66x50=3300

36[2]

S

S+1

S+2

S+9

S+10

S+11

S+18

S+19

S+20

S(S+20) = S2+20S

(S+18)(S+2) = S2+20S+36

The difference is 36

Now I am investigating 4x4 squares

Square

Products

Difference

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31

32

33

39

40

41

42

48

49

50

51

57

58

59

60

30x60=1800

57x33=1881

81

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32

33

34

4x34=136

31x7=217

81

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25

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34

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50

51

52

22x52=1144

49x25=1225

81

S

S+1

S+2

S+3

S+9

S+10

S+11

S+12

S+18

S+19

S+20

S+21

S+27

S+28

S+29

S+30

S(S+30) = S2+30S

(S+27)(S+3) = S2+30S+81

the difference is 81

My second theory (see Above†) will be proved right or wrong by my investigations into 5x5 squares within a 9x9 grid.

Square

Products

Difference

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16

21

22

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25

30

31

32

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34

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40

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49

50

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52

12x52=624

48x16=768

144

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81

41x81=3321

77x45=3465

144

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41

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48

49

50

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57

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68

28x68=1904

64x32=2048

144

S

S+1

S+2

S+3

S+4

S+9

S+10

S+11

S+12

S+13

S+18

S+19

S+20

S+21

S+22

S+27

S+28

S+29

S+30

S+31

S+36

S+37

S+38

S+39

S+40

S(S+40) = S2+S40

(S+36)(S+4) = S2+S40+144

This is the correct difference

My second theory (see Above†) was correct

I am now going to produce a formula for squares in a 9x9 grid

1

2

10

11

As before, I shall use my algebraic values to investigate 2x2, 3x3, 4x4 and 5x5 squares

                                                                   W

S

S+(w-1)

S+(9[w-1])

S+(9[w-1])+(w-1)

S(S+9W-9+W-1) = 1(11) =11

(S+9W-9)(S+W-1) = (10)(2)=20

the difference is 9, which is what it should be

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49

S

S+(W-1)

S+9(W-1)

S+9(W-1) +(W-1)

S(S+9[W-1]+[W-1]) = 29(49) = 1421

(S+9[W-1])(S+[W-1] = (47)(31) = 1457

the difference is 36, which is the correct difference

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52

...read more.

Conclusion

(4X5)

Rectangle

Products

Difference

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51X94=4794

91X54=4914

120

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6X49=294

46X9=414

120

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1X44=44

41X4=164

120

If I modify my formula for the difference of any size square in a 10X10 grid, I can create a formula for any size rectangle in a 10X10 grid.

I already have the terms:

  • S=the starting number
  • W=the width

Now I need one for the length I shall use O.

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45

To test my formula I shall use the rectangle

For this rectangle      

O=3

W=2

S=24

S

S+(W-1)

S+10(O-1)

S+10(O-1)+(W-1)

S(S+10[O-1]+[W-1]) = 24(45) = 1080

(S+10[O-1])(S+[W-1]) = (44)(25) = 1100

the difference is 20 which is the correct number for this rectangle

Page  of


[1]* I now have a theory ; I think that the number in the difference column in a table for a 2x2 grid is the same as the size of the grid the 2x2 square is in, I shall investigate this further after I have completed my investigation of a 9x9 grid

[2]† I have another theory, I think that the numbers in the difference column for a square in a 9x9 grid are 90% of the difference of the corresponding size squares in a 10x10 grid just as the size of a 9x9 grid is 90% of the size of a 10x10

[3]‡ I am extending my second theory to say that the number in the difference column of a certain size square in an 8x8 grid is 80% of the same number for the same size square in a 10x10 grid and 70% for a 7x7 grid etc.

...read more.

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